The cross-product conjecture for width two posets

Abstract

The cross--product conjecture (CPC) of Brightwell, Felsner and Trotter (1995) is a two-parameter quadratic inequality for the number of linear extensions of a poset $P=(X,\prec )$ with given value differences on three distinct elements in $X$. We give two different proofs of this inequality for posets of width two. The first proof is algebraic and generalizes CPC to a four-parameter family. The second proof is combinatorial and extends CPC to a $q$-analogue. Further applications include relationships between CPC and other poset inequalities, including a new $q$-analogue of the Kahn--Saks inequality.

31 pages, 7 figures. Counterexamples to Conjecture 11.2, 11.3, and 11.4 in v2 were found, and are now included in Section 11.5 and 11.10 of v3. To appear in Trans. Amer. Math. Soc

References in corpus (1)

• Extensions of the Kahn--Saks inequality for posets of width two

Cited by in corpus (5)

• Extensions of the Kahn--Saks inequality for posets of width two
• Multivariate correlation inequalities for $P$-partitions
• Log-concave poset inequalities
• Log-concavity in planar random walks
• Correlation inequalities for linear extensions